Chapter 8.4, Problem 3E

To determine

Find the partial sum for the given values of $n$ .

Answer to Problem 3E

The $S_n$ appears to approach $0.8$ as $n$ increases.

Explanation of Solution

Given:

The given infinite geometric series is $\frac{1}{2}+\frac{1}{6}+\frac{1}{18}+\frac{1}{54}+\frac{1}{162}+\cdot \cdot \cdot$ and the values of $n=1,2,3,4,\text{and}5$ .

Calculation:

The partial sums are

  $\begin{array}{l}{S}_1=\frac{1}{2}=0.5\\ S_2=\frac{1}{2}+\frac{1}{6}=0.6666\\ S_3=\frac{1}{2}+\frac{1}{6}+\frac{1}{18}=0.7222\\ S_4=\frac{1}{2}+\frac{1}{6}+\frac{1}{18}+\frac{1}{54}=0.740\\ S_5=\frac{1}{2}+\frac{1}{6}+\frac{1}{18}+\frac{1}{54}+\frac{1}{162}=0.7469\end{array}$

Plot the points $\left(1,0.5\right),\left(2,0.666\right),\left(3,0.7222\right),\left(4,0.740\right)\text{and}\left(5,0.7469\right)$ .

  

From the graph, $S_n$ appears to approach $0.8$ as $n$ increases.